Linear Algebra I would like the answers for subquestions a, b, c. Given that the matrix $A$ has the Singular Value Decomposition $A = U \Sigma V^T$, where:\[ U = \begin{pmatrix} \frac{2}{3} & \frac{1}{2} & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 & 0 \\ \frac{1}{3} & 0 & -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 & 0 \\ \frac{2}{3} & -\frac{1}{2} & 0 & 0 & \frac{\sqrt{3}}{6} & \frac{1}{6} \end{pmatrix}, \]\[ \Sigma = \begin{pmatrix} 6 & 0 & 0 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \]\[ V = \begin{pmatrix} \frac{1}{2} & 1 & 0 \\ \frac{1}{2} & -1 & 0 \\ -\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} \\ -\frac{1}{2} & 0 & -\frac{1}{\sqrt{2}}\end{pmatrix}\]Answer the questions below using the given SVD, without actually computing the entries in $A$. You should be able to answer all but the last part below without any calculation.(a) For this $m \times n$ matrix $A$, what is $m$ and what is $n$?(b) What is the **rank** $r$ of $A$?(c) Write down an orthonormal basis for $R(A)$.(d) Write down an orthonormal basis for $N(A)$.(e) Write down an orthonormal basis for $R(A^T)$.(f) Write down an orthonormal basis for $N(A^T)$.(g) Write down the **

Name: Linear Algebra I would like the answers for subquestions a, b, c. Give
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Description: Linear Algebra I would like the answers for subquestions a, b, c. Given that the matrix $A$ has the Singular Value Decomposition $A = U \Sigma V^T$, where:\[ U = \begin{pmatrix} \frac{2}{3} & \frac{1}{2} & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2}

The singular value decomposition of A is given as, A=UΣVT, where, U=2312-26130223-231226 ,…

Answered: Given that the matrix A has the…

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Chapter2: Second-order Linear Odes

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Linear Algebra

I would like the answers for subquestions a, b, c.

Given that the matrix $A$ has the Singular Value Decomposition $A = U \Sigma V^T$, where:

\[
U =
\begin{pmatrix}
\frac{2}{3} & \frac{1}{2} & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 & 0 \\
\frac{1}{3} & 0 & -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 & 0 \\
\frac{2}{3} & -\frac{1}{2} & 0 & 0 & \frac{\sqrt{3}}{6} & \frac{1}{6}
\end{pmatrix},
\]

\[
\Sigma =
\begin{pmatrix}
6 & 0 & 0 & 0 & 0 & 0 \\
0 & 5 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix},
\]

\[
V =
\begin{pmatrix}
\frac{1}{2} & 1 & 0 \\
\frac{1}{2} & -1 & 0 \\
-\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} \\
-\frac{1}{2} & 0 & -\frac{1}{\sqrt{2}}
\end{pmatrix}
\]

Answer the questions below using the given SVD, without actually computing the entries in $A$. You should be able to answer all but the last part below without any calculation.

(a) For this $m \times n$ matrix $A$, what is $m$ and what is $n$?

(b) What is the **rank** $r$ of $A$?

(c) Write down an orthonormal basis for $R(A)$.

(d) Write down an orthonormal basis for $N(A)$.

(e) Write down an orthonormal basis for $R(A^T)$.

(f) Write down an orthonormal basis for $N(A^T)$.

(g) Write down the **

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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