Final Exam Review Questions

Topic-by-topic practice

Reviews

Vectors, Linear Combinations, and Matrix Multiplication

Questions

  1. Let \[ u=\begin{pmatrix}1\\2\end{pmatrix}, \qquad v=\begin{pmatrix}-1\\3\end{pmatrix}. \] Compute \[ 2u-v. \]

  2. Write \[ \begin{pmatrix}5\\1\end{pmatrix} \] as a linear combination of \[ \begin{pmatrix}1\\1\end{pmatrix}, \qquad \begin{pmatrix}2\\-1\end{pmatrix}. \]

  3. Let \[ A=\begin{pmatrix}1&2\\0&-1\end{pmatrix}, \qquad x=\begin{pmatrix}3\\-2\end{pmatrix}. \] Compute \(Ax\).

  4. If \[ A=\begin{pmatrix}1&0&2\\-1&3&1\end{pmatrix}, \qquad B=\begin{pmatrix}1&2\\0&-1\\3&1\end{pmatrix}, \] what is the size of \(AB\)?

  5. If \(A\) is \(3\times 2\) and \(B\) is \(2\times 4\), what is the size of \(BA\)? If it does not exist, say so.

Solutions

  1. \[ 2u-v= \begin{pmatrix}3\\1\end{pmatrix}. \]

  2. \[ \begin{pmatrix}5\\1\end{pmatrix} = \frac73\begin{pmatrix}1\\1\end{pmatrix} + \frac43\begin{pmatrix}2\\-1\end{pmatrix}. \]

  3. \[ Ax= \begin{pmatrix}-1\\2\end{pmatrix}. \]

  4. The product has size \(2\times 2\).

  5. \(BA\) does not exist, because \(B\) is \(2\times 4\) and \(A\) is \(3\times 2\).

Linear Transformations, Systems, Inverses, Elementary Matrices

Questions

  1. A linear transformation satisfies \[ T(e_1)=\binom{2}{1}, \qquad T(e_2)=\binom{-1}{3}. \] Find \[ T\binom{2}{-1}. \]

  2. The RREF of an augmented matrix is \[ \left[ \begin{array}{ccc|c} 1&0&3&2\\ 0&1&-1&4\\ 0&0&0&0 \end{array} \right]. \] Does the system have one solution, no solution, or infinitely many solutions?

  3. For the system in Question 2, write the solution in parametric form.

  4. Let \[ A=\begin{pmatrix}1&2\\3&7\end{pmatrix}. \] Find \(A^{-1}\).

  5. True or false: Every elementary matrix is invertible.

  6. True or false: If a square matrix has an inverse, then its RREF is the identity matrix.

Solutions

  1. \[ T\binom{2}{-1} =\binom{5}{-1}. \]

  2. Infinitely many solutions.

  3. \[ (x_1,x_2,x_3)=(2-3t,\ 4+t,\ t). \]

  4. \[ A^{-1}= \begin{pmatrix} 7&-2\\ -3&1 \end{pmatrix}. \]

  5. True.

  6. True.

Markov Chains and Abstract Vector Spaces

Questions

  1. A two-state Markov chain has states L and S, with:

    • 80% of loyal customers stay loyal,
    • 30% of switchers become loyal.

    Write the transition matrix \(P\) using the convention \[ P_{ij}=\text{probability of moving from state }j\text{ to state }i. \]

  2. True or false: In this course convention, each row of a transition matrix sums to 1.

  3. In \(P_2\), determine whether \[ S=\{p(x)\mid p(1)=0\} \] is a subspace.

  4. In \(P_2\), determine whether \[ T=\{p(x)\mid p(1)=1\} \] is a subspace.

  5. In \(C(\mathbb R)\), determine whether \[ U=\{f\mid f(0)=0\} \] is a subspace.

Solutions

  1. \[ P= \begin{pmatrix} 0.8&0.3\\ 0.2&0.7 \end{pmatrix}. \]

  2. False. Each column sums to 1.

  3. Yes.

  4. No.

  5. Yes.

Linear Independence, Spanning Sets, Bases, Rank, Nullity

Questions

  1. Are the vectors \[ v_1=(1,0,1), \qquad v_2=(0,1,1), \qquad v_3=(1,1,2) \] linearly independent?

  2. Let \[ u=(1,2,0), \qquad v=(0,1,1), \qquad w=(1,3,1). \] Is \(w\) in \(\text{span}\{u,v\}\)?

  3. The RREF of a \(5\times 4\) matrix is \[ \begin{pmatrix} 1&0&2&0\\ 0&1&-1&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}. \] Find the rank and nullity.

  4. For the matrix in Question 3, write one basis vector for the nullspace.

  5. Let \[ V=\text{span}\left\{ \begin{pmatrix}1\\0\\1\end{pmatrix}, \begin{pmatrix}0\\1\\1\end{pmatrix}, \begin{pmatrix}1\\1\\2\end{pmatrix} \right\}. \] Find the dimension of \(V\).

  6. True or false: A basis is a spanning set that is linearly independent.

Solutions

  1. No, because \[ v_3=v_1+v_2. \]

  2. Yes, since \(w=u+v\).

  3. \[ \mathop{\mathrm{rank}}(A)=3, \qquad \mathop{\mathrm{null}}(A)=1. \]

  4. One basis vector is \[ \begin{pmatrix}-2\\1\\1\\0\end{pmatrix}. \]

  5. The dimension is \(2\).

  6. True.

Dot Products, Orthogonality, Projections, Least Squares

Questions

  1. Are the vectors \[ u=(1,2,1), \qquad v=(1,0,-1) \] orthogonal?

  2. Let \[ u=(1,2,2), \qquad v=(2,0,1). \] Compute \(\|u\|\).

  3. For the vectors in Question 2, compute \[ u\cdot v. \] Is the angle acute, right, or obtuse?

  4. Find the projection of \[ b=(3,1,2) \] onto \(\text{span}\{u\}\) where \[ u=(1,1,0). \]

  5. True or false: In least squares, the residual is orthogonal to the column space of \(A\).

  6. True or false: If two nonzero vectors are orthogonal, then they must be linearly dependent.

Solutions

  1. Yes, because the dot product is \(0\).

  2. \[ \|u\|=3. \]

  3. \[ u\cdot v=4, \] so the angle is acute.

  4. \[ \operatorname{proj}_u(b)=(2,2,0). \]

  5. True.

  6. False.

Determinants, Cramer’s Rule, Cross Product

Questions

  1. Compute \[ \det\begin{pmatrix}3&1\\2&5\end{pmatrix}. \]

  2. If \(\det(A)=2\) and \(\det(B)=-3\) for \(3\times 3\) matrices, compute \[ \det(ABA). \]

  3. Suppose for a \(3\times 3\) system \(Ax=b\), \[ \det(A)=5,\qquad \det(A_1)=10,\qquad \det(A_2)=-15,\qquad \det(A_3)=20. \] Find the solution using Cramer’s rule.

  4. Compute \[ (1,0,2)\times(0,1,1). \]

  5. If \(\det(A)=-4\), what happens to area and orientation in \(\mathbb R^2\) under the transformation \(x\mapsto Ax\)?

  6. True or false: If \(u\times v=0\), then \(u\) and \(v\) must be perpendicular.

Solutions

  1. \[ 13. \]

  2. \[ -12. \]

  3. \[ (x_1,x_2,x_3)=(2,-3,4). \]

  4. \[ (-2,-1,1). \]

  5. Area scales by \(4\) and orientation reverses.

  6. False.

Eigenvalues, Eigenvectors, Diagonalization, ODEs

Questions

  1. Find the eigenvalues of \[ A=\begin{pmatrix} 4&1\\ 0&2 \end{pmatrix}. \]

  2. Find an eigenvector of \[ A=\begin{pmatrix} 3&1\\ 0&2 \end{pmatrix} \] corresponding to \(\lambda=2\).

  3. A matrix \(A\) has eigenvectors \[ v_1=\binom{1}{0}, \qquad v_2=\binom{1}{1} \] with eigenvalues \(2\) and \(5\), respectively. Write down \(P\) and \(D\) such that \[ A=PDP^{-1}. \]

  4. Suppose \(A\) has eigenvectors \(u_1,u_2\) with eigenvalues \(5\) and \(-1\), respectively. Compute \[ A^3(2u_1-u_2). \]

  5. Solve \[ x'(t)=Dx(t), \qquad D=\begin{pmatrix}2&0\\0&-1\end{pmatrix}, \qquad x(0)=\binom{3}{1}. \]

  6. True or false: Every square matrix is diagonalizable.

  7. True or false: A \(90^\circ\) rotation matrix in \(\mathbb R^2\) has a real eigenvector.

Solutions

  1. The eigenvalues are \(4\) and \(2\).

  2. One eigenvector is \[ \binom{1}{-1}. \]

  3. \[ P=\begin{pmatrix}1&1\\0&1\end{pmatrix}, \qquad D=\begin{pmatrix}2&0\\0&5\end{pmatrix}. \]

  4. \[ 250u_1+u_2. \]

  5. \[ x(t)=\binom{3e^{2t}}{e^{-t}}. \]

  6. False.

  7. False.

SVD and PCA

Questions

  1. True or false: Every real matrix has an SVD.

  2. True or false: Singular values can be negative.

  3. A matrix has singular values \[ 8,\ 3,\ 0,\ 0. \] If it has 4 columns, what are its rank and nullity?

  4. If a matrix has singular values \(20,5,1,0\), what singular value would a rank-1 approximation keep?

  5. In PCA, suppose the first principal direction is \[ v_1=\frac{1}{\sqrt2}(1,1,0). \] Which combination of coordinates has the largest variance?

  6. Explain briefly how SVD is similar to diagonalization and how it is different.

Solutions

  1. True.

  2. False.

  3. \[ \mathop{\mathrm{rank}}(A)=2, \qquad \mathop{\mathrm{null}}(A)=2. \]

  4. The rank-1 approximation keeps \[ 20. \]

  5. The equal-weight combination of the first two coordinates.

  6. Both describe important directions of a matrix. Diagonalization may fail to exist, but SVD always exists.

Mixed True / False Concept Check

Questions

  1. True or false: A linear transformation is determined by its values on a basis.
  2. True or false: If \(Ax=0\) has a nontrivial solution, then the columns of \(A\) are linearly independent.
  3. True or false: Every elementary matrix is invertible.
  4. True or false: In this course convention for Markov chains, each column of \(P\) sums to 1.
  5. True or false: The set \(\{p\in P_2\mid p(1)=1\}\) is a subspace.
  6. True or false: If \(\det(A)=0\), then \(A\) is not invertible.
  7. True or false: Eigenvectors corresponding to distinct eigenvalues are linearly independent.
  8. True or false: The number of nonzero singular values equals the rank.

Solutions

  1. True.
  2. False.
  3. True.
  4. True.
  5. False.
  6. True.
  7. True.
  8. True.

Longer Concept Questions

Questions

  1. Let \[ A= \begin{pmatrix} 2&-1\\ 1&2 \end{pmatrix} \] and suppose \(\det(A)=5\). Explain what the columns mean geometrically, what happens to the unit square, and why \(A\) is invertible.

  2. Consider the set \[ S=\left\{ \begin{pmatrix} a+b\\ a-b\\ 2a \end{pmatrix} : a,b\in\mathbb R \right\}. \] Explain why \(S\) is a subspace, find a basis, and state its dimension.

  3. Suppose the RREF of an augmented matrix is \[ \left[ \begin{array}{cccc|c} 1&0&2&0&1\\ 0&1&-1&3&0\\ 0&0&0&0&0 \end{array} \right]. \] Describe the solution set geometrically and explain what rank and nullity tell you.

  4. Let \[ u_1=\frac{1}{\sqrt2}(1,1,0), \qquad u_2=\frac{1}{\sqrt2}(1,-1,0), \qquad b=(3,1,2). \] Find the projection of \(b\) onto \(\text{span}\{u_1,u_2\}\) and explain why it is the best approximation.

  5. Suppose \(A\) has eigenvectors \(u_1,u_2\) with eigenvalues \(1.2\) and \(0.6\), and \[ x_0=5u_1+2u_2. \] Describe the long-term behavior of \(A^k x_0\).

  6. A matrix has singular values \[ 20,\ 5,\ 1,\ 0. \] Explain what a rank-1 approximation keeps and why the first singular vector matters in PCA.

Solution Sketches

  1. The columns are the images of \(e_1\) and \(e_2\). The unit square becomes the parallelogram spanned by those columns. Since \(\det(A)=5\), area scales by \(5\) and \(A\) is invertible.

  2. Since every vector in \(S\) is \[ a\begin{pmatrix}1\\1\\2\end{pmatrix} + b\begin{pmatrix}1\\-1\\0\end{pmatrix}, \] the set is a span, hence a subspace. Those two vectors form a basis, so the dimension is \(2\).

  3. The solution set is an affine plane in \(\mathbb R^4\). There are 2 pivot variables and 2 free variables, so rank is \(2\) and nullity is \(2\).

  4. The projection is \[ (3,1,0), \] and the residual \[ (0,0,2) \] is orthogonal to the subspace, which is why the approximation is best.

  5. The term in the \(u_1\) direction dominates because \((1.2)^k\) grows while \((0.6)^k\) decays, so the vector eventually points mostly in the \(u_1\) direction.

  6. A rank-1 approximation keeps only the largest singular value and its singular vectors, so it captures the strongest pattern in the data. The first singular vector gives the direction of largest variance, which is the first principal component.

Concept Reminders

Short conceptual answers

  • What does a column of a matrix mean? It is the image of a basis vector under the linear transformation.
  • What is the difference between span and basis? A basis is a spanning set that is also linearly independent.
  • What does rank measure? It measures how many independent columns or pivot directions the matrix has.
  • What does the determinant measure geometrically? It measures signed area or volume scaling.
  • What does the nullspace tell you? It tells you which inputs are sent to zero.
  • Why is the least-squares residual orthogonal to the column space? Because the best approximation occurs when the error has no component in the model space.
  • Why are eigenvalues useful? They reveal directions where the transformation acts by pure scaling.
  • What is the big idea of diagonalization? Change to an eigenbasis so powers and exponentials become easy to compute.
  • What do singular values tell you? They tell you how much the matrix stretches orthogonal directions.
  • What is PCA trying to do? It keeps the directions of largest variance while reducing dimension.