91Ó°ÊÓ

Let \(A\) be a \(4 \times 5\) matrix and let \(U\) be the reduced row echelon form of \(A\). If \\[ \begin{array}{l} \mathbf{a}_{1}=\left(\begin{array}{r} 2 \\ 1 \\ -3 \\ -2 \end{array}\right), \quad \mathbf{a}_{2}=\left(\begin{array}{rrr} -1 \\ 2 \\ 3 \\ 1 \end{array}\right) \\ U=\left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -1 \\ 0 & 1 & 3 & 0 & -2 \\ 0 & 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right) \end{array} \\] (a) find a basis for \(N(A)\) (b) given that \(\mathbf{x}_{0}\) is a solution to \(A \mathbf{x}=\mathbf{b},\) where \\[ \mathbf{b}=\left(\begin{array}{l} 0 \\ 5 \\ 3 \\ 4 \end{array}\right) \text { and } \mathbf{x}_{0}=\left(\begin{array}{l} 3 \\ 2 \\ 0 \\ 2 \\ 0 \end{array}\right) \\] (i) \(\quad\) find all solutions to the system. (ii) determine the remaining column vectors of \(A\).

Let \(S\) be the subspace of \(P_{3}\) consisting of all polynomials \(p(x)\) such that \(p(0)=0,\) and let \(T\) be the subspace of all polynomials \(q(x)\) such that \(q(1)=\) 0\. Find bases for (a) \(S\) (b) \(T\) (c) \(S \cap T\)

Let \(S\) denote the set of all infinite sequences of real numbers with scalar multiplication and addition defined by \\[ \begin{aligned} \alpha\left\\{a_{n}\right\\} &=\left\\{\alpha a_{n}\right\\} \\ \left\\{a_{n}\right\\}+\left\\{b_{n}\right\\} &=\left\\{a_{n}+b_{n}\right\\} \end{aligned} \\] Show that \(S\) is a vector space.

Let \(A\) be a \(5 \times 8\) matrix with rank equal to 5 and let b be any vector in \(\mathbb{R}^{5}\). Explain why the system \(A \mathbf{x}=\mathbf{b}\) must have infinitely many solutions.

Let \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{k}\right\\}\) be a spanning set for a vector space \(V\) (a) If we add another vector, \(\mathbf{x}_{k+1},\) to the set, will we still have a spanning set? Explain. (b) If we delete one of the vectors, say, \(\mathbf{x}_{k}\), from the set, will we still have a spanning set? Explain.

Let \(A\) be a \(5 \times 3\) matrix of rank 3 and let \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}\right\\}\) be a basis for \(\mathbb{R}^{3}\) (a) Show that \(N(A)=\\{0\\}\) (b) Show that if \(\mathbf{y}_{1}=A \mathbf{x}_{1}, \mathbf{y}_{2}=A \mathbf{x}_{2},\) and \(\mathbf{y}_{3}=A \mathbf{x}_{3}\) then \(\mathbf{y}_{1}, \mathbf{y}_{2},\) and \(\mathbf{y}_{3}\) are linearly independent. (c) Do the vectors \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\) from part \((\mathbf{b})\) form a basis for \(\mathbb{R}^{5}\) ? Explain.

Prove that a linear system \(A \mathbf{x}=\mathbf{b}\) is consistent if and only if the rank of \((A | \mathbf{b})\) equals the rank of \(A\).

Prove that if \(S\) is a subspace of \(\mathbb{R}^{1},\) then either \(S=\\{0\\}\) or \(S=\mathbb{R}^{1}\).

Prove that if \(S\) is a subspace of \(\mathbb{R}^{1},\) then either \(S=\\{\mathbf{0}\\}\) or \(S=\mathbb{R}^{1}\).

Let \(S\) be the subspace of \(\mathbb{R}^{2}\) spanned by \(\mathbf{e}_{1}\) and let \(T\) be the subspace of \(\mathbb{R}^{2}\) spanned by \(\mathbf{e}_{2}\). Is \(S \cup T\) a subspace of \(\mathbb{R}^{2} ?\) Explain.