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• Basis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let \[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]
• Find All Values of $x$ such that the Matrix is Invertible Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if \[ A= \begin{bmatrix} 1 & 0 & c \\ 0 & a & -b \\ -1/a & x & x^{2} \end{bmatrix} . \]   Solution. We know that $A$ is invertible precisely when […]
• Find All Eigenvalues and Corresponding Eigenvectors for the $3\times 3$ matrix Find all eigenvalues and corresponding eigenvectors for the matrix $A$ if \[ A= \begin{bmatrix} 2 & -3 & 0 \\ 2 & -5 & 0 \\ 0 & 0 & 3 \end{bmatrix} . \]   Solution. If $\lambda$ is an eigenvalue of $A$, then $\lambda$ […]
• Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3. Let $A$ be the matrix given by \[ A= \begin{bmatrix} -2 & 0 & 1 \\ -5 & 3 & a \\ 4 & -2 & -1 \end{bmatrix} \] for some variable $a$. Find all values of $a$ which will guarantee that $A$ has eigenvalues $0$, $3$, and $-3$.   Solution. Let $p(t)$ be the […]
• Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.   Proof. To prove that $W$ is a subspace of $V$, the $3\times 3$ zero matrix […]
• In which $\R^k$, are the Nullspace and Range Subspaces? Let $A$ be an $m \times n$ matrix. Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.   Solution. For an $m \times n$ matrix $A$, the […]
• Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors Find a basis for $\Span(S)$ where $S= \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} , \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix} , \begin{bmatrix} 2 \\ 6 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}$.   Solution. We […]
• Find a Basis for the Subspace spanned by Five Vectors Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where \[ \mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 […]