1

Let \((x_1,y_1)\) and \((x_2,y_2)\) both be solutions of the equation

1. Prove that \((x_1+x_2,y_1+y_2)\) is also a solution of (4.1.1).
2. If \(c\in \mathbb{R}\) is any real number, prove that \((cx_1,cy_1)\) is also a solution of (4.1.1).

2

Let \((x_1,y_1)\) and \((x_2,y_2)\) both be solutions of the equation

1. Prove that \((x_1+x_2,y_1+y_2)\) is not a solution of (4.1.2).
2. Assume \(c\in \mathbb{R}\) is a real number. Prove that if \((cx_1,cy_1)\) is a solution of (4.1.2), then we must have \(c=1\text{.}\)
3. Comparing this problem with the previous, explain: What is the most important difference between (4.1.1) and (4.1.2) that gives rise to these different conclusions? Why?

3

Consider the equation

1. Let \((x_1,y_1)=(-4,0)\) and \((x_2,y_2) = (2,12)\text{.}\) Prove that \((x_1,y_1)\) and \((x_2,y_2)\) are both solutions of (4.1.3). Also, prove that \((x_1+x_2,y_1+y_2)\) is a solution of (4.1.3).
2. Prove or disprove: For all \(x_1,x_2,y_1,y_2\text{,}\) if \((x_1,y_1)\) and \((x_2,y_2)\) are both solutions of (4.1.3), then \((x_1+x_2,y_1+y_2)\) is a solution of (4.1.3).

4

Supply an argument in favor of the following statement: The equation

is a linear equation.

5

Prove by examples that a linear system of 4 equations and 3 unknowns may have any of the following:

1. No solution.
2. A unique solution.
3. Infinitely many solutions.

6

Prove or disprove that a linear system of 3 equations and 4 unknowns may have each of the following:

1. No solution.
2. A unique solution.
3. Infinitely many solutions.

7

Prove that if a linear system of equations has a unique solution, then it has no free variables.

8

Consider a linear system of \(m\) equations and \(n\) unknowns.

1. Complete the sentence, and provide a convincing explanation: If this system has \(b\) basic variables, then \begin{equation*} b \leq \underline{\qquad}. \end{equation*}
2. Complete the sentence, and provide a convincing explanation: If this system has \(f\) free variables, then \begin{equation*} f = \underline{\qquad}. \end{equation*} (Feel 'free' to use some of \(m,n,\) and/or \(b\) in your equation.)
3. Prove that, if \(m\gt n\text{,}\) then this system cannot have a unique solution.

9

Let \({\bf v_1}, {\bf v_2}, {\bf v_3}\) be arbitrary vectors in \(\mathbb{R}^3\text{.}\)

1. Prove that there is always a linear combination of \({\bf v}_1, {\bf v}_2, {\bf v}_3\) that is equal to the zero vector \({\bf 0} = (0,0,0)\text{.}\)
2. Assume that there is more than one different linear combination that satisfies part (a). Prove that in this case, there must be infinitely many linear combinations of \({\bf v}_1, {\bf v}_2, {\bf v}_3\) that equal the zero vector.
3. True or false, and explain: These results will still hold if the three vectors live in \(\mathbb{R}^n\) for any \(n \geq 1\text{.}\)

10

Suppose \({\bf v}_1,{\bf v}_2,\ldots,{\bf v}_n \in \mathbb{R}^m\) is a collection of \(m\)-dimensional vectors, and suppose that the matrix

has a pivot position in every row.

1. Let \({\bf b} \in \mathbb{R}^m\) be an arbitrary \(m\)-dimensional vector. Prove that \({\bf b}\) can be written as a linear combination of the vectors \({\bf v}_1,{\bf v}_2,\ldots,{\bf v}_n\text{.}\)
2. Prove that, if \(m\gt n\text{,}\) the conclusion of part (a) is false.

11

Let \(A\) be an \(m\times n\) matrix and \({\bf b} \in \mathbb{R}^m\) be a vector.

1. Prove that if \({\bf x}_1\) and \({\bf x}_2\) are both solutions of the equation \(A{\bf x}={\bf b}\text{,}\) then the vector \begin{equation*} {\bf v} = {\bf x}_1-{\bf x}_2 \end{equation*} is a solution of the equation \(A{\bf x} = {\bf 0}.\)
2. Assume that the equation \(A{\bf x} = {\bf b}\) has a unique solution. Use part (a) to determine the solution set of the equation \(A{\bf x} = {\bf 0}.\)
3. Conversely, suppose \({\bf x}\) is a vector satisfying \(A{\bf x} = {\bf b}\text{,}\) and \({\bf n} \neq {\bf 0}\) is a nonzero vector such that \(A{\bf n} = {\bf 0}.\) Prove that \begin{equation*} {\bf v} = {\bf x} + {\bf n} \end{equation*} also satisfies the equation \(A{\bf x} = {\bf b}.\)

12

Let \(A\) be an \(n\times n\) square matrix. Suppose that \({\bf v}_1, {\bf v}_2 \in {\mathbb R}^n\) are vectors which have the property that

Let \({\bf x} \in {\rm span}\{{\bf v}_1,{\bf v}_2\}.\) Prove that, in the limit as \(n\to\infty\text{,}\)

13

Let \(A\) be an arbitrary \(3\times 4\) matrix and \(B\) be an arbitrary \(4\times 5\) matrix. The dimensions of these matrices make it possible to form the product matrix \(P = AB\text{.}\)

1. What are the dimensions of the product matrix \(P\text{?}\)
2. Prove that if the columns of \(A\) span \(\mathbb{R}^3\) and the columns of \(B\) span \(\mathbb{R}^4\text{,}\) then we can guarantee the columns of \(P\) span \(\mathbb{R}^3\text{.}\)

14

Let \({\bf v}_1,{\bf v}_2,{\bf w}\) be a set of vectors that is linearly dependent.

1. Prove that \({\bf w}\) can be written as a linear combination of \({\bf v}_1,{\bf v}_2\text{.}\)
2. Let \(S = {\rm span}\{{\bf v}_1,{\bf v}_2,{\bf w}\}\) and \(T = {\rm span}\{{\bf v}_1,{\bf v}_2\}.\) Prove directly from the definition of span that, as sets, \(S = T\text{.}\)

15

Let \(A\) be an \(n\times n\) square matrix, and \({\bf e}_1,{\bf e}_2,\ldots,{\bf e}_n\) be the standard vectors in \(\mathbb{R}^n.\)

Prove that if \(A{\bf e}_1, A{\bf e}_2,\ldots, A{\bf e}_n\) are linearly independent, then the equation

has a unique solution for all \({\bf b}.\)

16

Let \(A\) be a \(3\times 4\) matrix and \(B\) be a \(4\times 5\) matrix. Suppose that the product

is a matrix whose entries are all zero.

1. Prove that either (or both) matrix \(A\) and/or \(B\) must be "missing a pivot," that is, has at least one row without a pivot.
2. For the matrix \begin{equation*} A = \left[ \begin{array}{rrrr} 1\amp 0 \amp -1 \amp 1\\ 0\amp 1\amp 1\amp -2\\ 1\amp -1\amp 2\amp 0\end{array}\right]\text{,} \end{equation*} find a (nonzero!) \(4\times 5\) matrix \(B\) for which \(AB = O\text{.}\)