# SOLVED: 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

202.5a: Fundamental Theorem of Linear Algebra
202.5a: Fundamental Theorem of Linear Algebra

Get 5 free video unlocks on our app with code GOMOBILE

Snapsolve any problem by taking a picture.
Try it in the Numerade app?

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).

This problem has been solved!

Try Numerade free for 7 days

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) $B=\left\{E_{21}, E_{11}, E_{22}, E_{12}\right\} ; C=\left\{x, 1, x^{3}, x^{2}\right\}$

Oops! There was an issue generating an instant solution

An exclusive promotional rate that’s yours to keep forever

or

EMAIL