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Linear Transformation problems and solutions


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  • Projection to the subspace spanned by a vectorProjection to the subspace spanned by a vector Let $T: \R^3 \to \R^3$ be the linear transformation given by orthogonal projection to the line spanned by $\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}$. (a) Find a formula for $T(\mathbf{x})$ for $\mathbf{x}\in \R^3$. (b) Find a basis for the image subspace of $T$. (c) Find […]
  • Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$ Let \[A=\begin{bmatrix} 1 & -14 & 4 \\ -1 &6 &-2 \\ -2 & 24 & -7 \end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix} 4 \\ -1 \\ -7 \end{bmatrix}.\] Find $A^{10}\mathbf{v}$. You may use the following information without proving […]
  • The Preimage of a Normal Subgroup Under a Group Homomorphism is NormalThe Preimage of a Normal Subgroup Under a Group Homomorphism is Normal Let $G$ and $G'$ be groups and let $f:G \to G'$ be a group homomorphism. If $H'$ is a normal subgroup of the group $G'$, then show that $H=f^{-1}(H')$ is a normal subgroup of the group $G$.   Proof. We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup […]
  • A Square Root Matrix of a Symmetric MatrixA Square Root Matrix of a Symmetric Matrix Answer the following two questions with justification. (a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix. (b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]
  • Problems and Solutions About Similar MatricesProblems and Solutions About Similar Matrices Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix. Prove the following statements. (a) If $A$ is similar to $B$, then $B$ is similar to $A$. (b) $A$ is similar to itself. (c) If $A$ is similar to $B$ and $B$ […]
  • A Group Homomorphism is Injective if and only if MonicA Group Homomorphism is Injective if and only if Monic Let $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$. Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […]
  • Quiz 13 (Part 2) Find Eigenvalues and Eigenvectors of a Special MatrixQuiz 13 (Part 2) Find Eigenvalues and Eigenvectors of a Special Matrix Find all eigenvalues of the matrix \[A=\begin{bmatrix} 0 & i & i & i \\ i &0 & i & i \\ i & i & 0 & i \\ i & i & i & 0 \end{bmatrix},\] where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric […]
  • Determine Whether the Following Matrix Invertible. If So Find  Its Inverse Matrix.Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix. Let A be the matrix \[\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.\] Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse. (The Ohio State University Linear Algebra […]

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