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Then by the Fundamental Theorem of Linear Algebra: 7 â‰¤ rank(L) â‰¤ min(dim(L), dim(LT)). With A = null(L) and B = range(L), the matrix representation of L, relative to the decomposition of R^3 = A âŠ• B, is given by:

1. A is a k x k matrix.

2. B is a (-k) x (-k) matrix.

3. D is a (-W) x (-k) matrix.

4. k = rank(L).

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00:50

Diagonalize this matrix by constructing its eigenvalue matrix $\Lambda$ and its eigenvector matrix $S$ :$$A=\left[\begin{array}{cc}2 & 1-i \\1+i & 3\end{array}\right]=A^{\mathrm{H}}$$

03:16

For a set of column vectors (v1, v2, …, vn), we form matrix A = [v1 v2 … vn]. The reduced row echelon form for the matrix A is R = [1 3 … D; 0 0 … 0; …; 0 0 … 0], where D is a constant. Find D such that v1 + b*v2 + c*v3 = 0.

00:52

Determine the linear transformation $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ that has the given matrix.$$A=\left[\begin{array}{rr} 1 & 3 \\ -4 & 7 \end{array}\right]$$.

04:08

Determine the matrix representation $[T]_{B}^{C}$ for the given linear transformation $T$ and ordered bases $\bar{B}$ and $C$.$T: M_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R})$ given by$$T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=(a-d)+3 b x^{2}+(c-a) x^{3}$$(a) $B=\left\{E_{11}, E_{12}, E_{21}, E_{22}\right\} ; C=\left\{1, x, x^{2}, x^{3}\right\}$(b) $B=\left\{E_{21}, E_{11}, E_{22}, E_{12}\right\} ; C=\left\{x, 1, x^{3}, x^{2}\right\}$

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