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Example 1

Let \(p_1 = -x+1\), \(p_2 = x+2\), and \(p_3 = x^2+1\). Show that \(\{p_1,p_2,p_3\}\) is a basis for \(P_2\), the vector space of polynomials in \(x\) with real coefficients having degree at most \(2\). Write \(x^2 + 2x\) as a linear combination of \(p_1,p_2,p_3\).  

Example 2

Let \(A = \begin{bmatrix} 1 & 1\\ 0 & 2 \\ -1 & 0 \end{bmatrix}\) be a matrix with real entries. Is \(\begin{bmatrix} 1\\1\\1\end{bmatrix}\) in the column space of \(A\)?  

Example 3

Let \(A = \begin{bmatrix} 1 & -1 & 1 & 0\\ 0 & 1 & 1 & 0\\ -2 & 2 & 0 & 1 \end{bmatrix}\). Give a basis for each of \(N(A)\), \({\cal C}(A)\), and \({\cal R}(A)\).  

Example 4

Let \(A \in \mathbb{R}^{2\times 4}\). What is the smallest possible value for the nullity of \(A\)?  

Example 5

Let \(A = \begin{bmatrix} 1 & -2 & 1 \\ -2 & 4 & -2 \\ \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 2 & 1\\ 1 & 1 & 0\\ 1 & 0 & -1 \end{bmatrix}\) be matrices defined over the real numbers. Determine if \(N(A) = {\cal C}(B)\).  

Example 6

Let \(A = \begin{bmatrix} 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 1 & 0 & 1\\ 0 & 1 & 0 & 1 & 1\\ \end{bmatrix}\) be defined over \(GF(2)\). Give a basis for \(N(A)\).  

Example 7

Let \(\mathbb{F}\) denote a field. Let \(m\) and \(n\) be positive integers. Let \(A \in \mathbb{F}^{m \times n}\). Let \(b \in \mathbb{F}^m\). Show that \(b \in {\cal C}(A)\) if and only if the system of linear equations \(Ax = b\) has a solution where \(x =\begin{bmatrix} x_1\\ \vdots \\ x_n\end{bmatrix}\).  

Example 8

Let \(\Gamma = \left ( \begin{bmatrix} 1 \\ 2\end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}\right )\) be an ordered basis for \(\mathbb{R}^2\). Let \(u = \begin{bmatrix} 0 \\ -2 \end{bmatrix}\). What is \([u]_{\Gamma}\)?  

Example 9

Let \(\Gamma = ( x^2-1, x+1, x)\) be an ordered basis for the vector space of polynomials in \(x\) with real coefficients having degree at most \(2\). Let \(u = x-1\). What is \([u]_{\Gamma}\)?  

Example 10

Determine the dimension of the subspace of \(\mathbb{R}^{2\times 2}\) given by \(\left \{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : \begin{array}{r} a + b + c + d = 0 \\~b - c + 2d= 0 \end{array} \right \}\).  

Example 11

Let \(t,u,v,w \in \mathbb{R}^3\) be given by \(t = \begin{bmatrix} 1 \\ 0 \\ -2\end{bmatrix}\), \(u = \begin{bmatrix} 1 \\ -1\\ 2\end{bmatrix}\), \(v = \begin{bmatrix} -3 \\ 2 \\ -2\end{bmatrix}\), \(w = \begin{bmatrix} 2 \\ -1 \\ 0\end{bmatrix}\). Let \(W\) be a subspace of \(\mathbb{R}^3\) given by the span of \(\{t,u,v,w\}\). Show that \(W\) is a proper subspace of \(\mathbb{R}^3\).  

Example 12

Give a procedure for finding an orthogonal basis for \(\mathbb{R}^n\) starting with a given nonzero vector.