Practice Test 1 — Chapter 1

100 points · 5 extra credit

Practice Test 1 — Linear Equations in Linear Algebra (Chapter 1)

Practice for Test 1 · 100 points core + 5 points extra credit · time 90 minutes. No calculator. Show all reasoning; partial credit is given. Justify every "yes/no" answer with a theorem, a computation, or a counterexample.

1. 1. Row Reduction (12 points). Consider the system

   $\begin{aligned} x_1 - 3x_2 + 2x_3 - x_4 &= 2 \\ 3x_1 - 9x_2 + 7x_3 - x_4 &= 8 \\ -2x_1 + 6x_2 - 3x_3 + 4x_4 &= -2 \end{aligned}$

   (a) (6 pts) Write the augmented matrix and row reduce it to reduced echelon form.

   (b) (2 pts) Identify the pivot columns, the basic variables, and the free variables.

   (c) (4 pts) Write the general solution, solving for the basic variables in terms of the free variables.

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2. 2. Existence & Uniqueness (8 points). Consider the system

   $\begin{aligned} x_1 - x_2 + 2x_3 &= 3 \\ 3x_1 - 2x_2 + 5x_3 &= 7 \\ 2x_1 + x_2 + hx_3 &= k \end{aligned}$

   Find all values of $h$ and $k$ for which the system has

   (a) (3 pts) a unique solution;

   (b) (3 pts) no solution;

   (c) (2 pts) infinitely many solutions.

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3. 3. Span (10 points). Let $\mathbf a_1=\begin{bmatrix}1\\-2\\3\end{bmatrix}$, $\mathbf a_2=\begin{bmatrix}3\\-5\\7\end{bmatrix}$, $\mathbf a_3=\begin{bmatrix}-1\\4\\-7\end{bmatrix}$, and let $A=[\,\mathbf a_1\ \mathbf a_2\ \mathbf a_3\,]$.

   (a) (5 pts) Find the condition on $b_1,b_2,b_3$ for $\mathbf b=\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$ to lie in $\operatorname{Span}\{\mathbf a_1,\mathbf a_2,\mathbf a_3\}$.

   (b) (3 pts) Is $\mathbf b=\begin{bmatrix}2\\-3\\4\end{bmatrix}$ in $\operatorname{Span}\{\mathbf a_1,\mathbf a_2,\mathbf a_3\}$? If so, write $\mathbf b$ as a linear combination of $\mathbf a_1,\mathbf a_2,\mathbf a_3$.

   (c) (2 pts) Do the columns of $A$ span $\mathbb R^3$? Justify.

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4. 4. Solution Sets (12 points). Let $A=\begin{bmatrix}1&-2&1&2\\3&-6&4&9\\-2&4&-1&-1\end{bmatrix}$ and $\mathbf b=\begin{bmatrix}1\\5\\0\end{bmatrix}$.

   (a) (5 pts) Write the solution set of $A\mathbf x=\mathbf 0$ in parametric vector form.

   (b) (5 pts) Write the solution set of $A\mathbf x=\mathbf b$ in parametric vector form.

   (c) (2 pts) Describe both solution sets geometrically and say how they are related.

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5. 5. Linear Independence (10 points). Let $\mathbf v_1=\begin{bmatrix}1\\2\\-1\end{bmatrix}$, $\mathbf v_2=\begin{bmatrix}3\\7\\-2\end{bmatrix}$, $\mathbf v_3=\begin{bmatrix}-1\\h\\4\end{bmatrix}$.

   (a) (5 pts) Find all values of $h$ for which $\{\mathbf v_1,\mathbf v_2,\mathbf v_3\}$ is linearly dependent.

   (b) (5 pts) For that value of $h$, find a linear dependence relation among $\mathbf v_1,\mathbf v_2,\mathbf v_3$ with integer weights.

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6. 6. Balancing a Chemical Equation (6 points). Ethane burns in oxygen:

   $x_1\,\mathrm{C_2H_6} + x_2\,\mathrm{O_2} \;\to\; x_3\,\mathrm{CO_2} + x_4\,\mathrm{H_2O}$

   Set up the vector equation (one vector per molecule, counting C, H, O), solve the resulting homogeneous system, and give the balanced equation with the smallest positive whole-number coefficients.

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7. 7. The Matrix of a Linear Transformation (12 points).

   (a) (5 pts) Let $T:\mathbb R^2\to\mathbb R^2$ first rotate points clockwise through $\pi/2$ radians and then reflect them through the line $x_2=x_1$. Find the standard matrix of $T$.

   (b) (2 pts) $T$ is a single familiar geometric transformation. Which one?

   (c) (5 pts) Let $S:\mathbb R^2\to\mathbb R^3$ be a linear transformation with $S\!\left(\begin{bmatrix}1\\2\end{bmatrix}\right)=\begin{bmatrix}3\\0\\5\end{bmatrix}$ and $S\!\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\begin{bmatrix}1\\-1\\2\end{bmatrix}$. Find the standard matrix of $S$.

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8. 8. One-to-One & Onto (10 points). Let $T:\mathbb R^3\to\mathbb R^4$ be defined by

   $T(x_1,x_2,x_3)=(x_1+2x_2,\;\;-x_1+x_2+3x_3,\;\;2x_1+x_2-2x_3,\;\;3x_2+4x_3).$

   (a) (2 pts) Find the standard matrix $A$ of $T$.

   (b) (4 pts) Is $T$ one-to-one? Justify.

   (c) (4 pts) Does $T$ map $\mathbb R^3$ onto $\mathbb R^4$? Justify.

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9. 9. True or False (12 points, 2 each). State whether each statement is true or false, and justify: cite a theorem or definition if true, give a counterexample if false.

   (a) If an augmented matrix has a pivot in every row, then the system it represents is consistent.

   (b) If $A$ is a $3\times3$ matrix and $A\mathbf x=\mathbf 0$ has only the trivial solution, then the columns of $A$ span $\mathbb R^3$.

   (c) If $A$ is a $4\times3$ matrix, then $A\mathbf x=\mathbf b$ has a solution for every $\mathbf b$ in $\mathbb R^4$.

   (d) The solution set of $A\mathbf x=\mathbf b$ is always a translate of the solution set of $A\mathbf x=\mathbf 0$.

   (e) If $T$ is linear and $T(\mathbf u)=T(\mathbf v)$ for some $\mathbf u\neq\mathbf v$, then $T$ is not one-to-one.

   (f) Any set of vectors in $\mathbb R^n$ that contains the zero vector is linearly dependent.

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10. 10. Proofs (8 points).

    (a) (4 pts) Let $A$ be an $m\times n$ matrix. Prove that if $\mathbf p$ and $\mathbf q$ are both solutions of $A\mathbf x=\mathbf b$, then $\mathbf p-\mathbf q$ is a solution of $A\mathbf x=\mathbf 0$.

    (b) (4 pts) Let $T:\mathbb R^n\to\mathbb R^m$ be a one-to-one linear transformation. Prove that if $\{\mathbf v_1,\mathbf v_2,\mathbf v_3\}$ is linearly independent, then $\{T(\mathbf v_1),T(\mathbf v_2),T(\mathbf v_3)\}$ is linearly independent.

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11. Extra Credit (5 points ••)

    Let $A$ be an $m\times n$ matrix. Prove that if $A\mathbf x=\mathbf b$ has exactly one solution for some $\mathbf b$, then $A\mathbf x=\mathbf c$ has at most one solution for every $\mathbf c$ in $\mathbb R^m$.

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