Suggested languages for you:

Americas

Europe

Q7E

Expert-verifiedFound in: Page 176

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Which of the subsets V ****of ${{\mathbf{\mathbb{R}}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$**** given in Exercise 6 ****through 11**** are subspaces of ${{\mathbf{\mathbb{R}}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$****. The diagonal 3x3**** matrices.**

The diagonal 3x3 matrices subset V of ${\mathrm{\mathbb{R}}}^{3x3}$ is a subspace of ${\mathrm{\mathbb{R}}}^{3x3}$.

**A subset W**** of a linear space V**** is called a subspace of V**** if **

**(a) W contains the neutral element 0**** of V****.**

**(b) W is closed under addition (if f**** and g**** are in W**** then so is f+g ****)**

**(c) W is closed under scalar multiplication (if f ****is in W**** and k**** is scalar, then kf**** is in W****). **

we can summarize parts b and c by saying that W is closed under linear combinations.

Consider two 3x3 matrix namely,

$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}B=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]\phantom{\rule{0ex}{0ex}}\left|A\right|=1\left(1-0\right)-0+0\phantom{\rule{0ex}{0ex}}\left|A\right|=1$

Find A+B.

$A+B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]+\left[\begin{array}{ccc}4& 0& 0\\ 0& -3& 0\\ 0& 0& -1\end{array}\right]\phantom{\rule{0ex}{0ex}}A+B=\left[\begin{array}{ccc}4& 0& 0\\ 0& -1& 0\\ 0& 0& 2\end{array}\right]$

Thus, the sum of two diagonal matrices is a diagonal matrix, because all other entries will be zero and only the entries in the diagonal can be added and remain non-zero elements.

Similarly, when multiplying a matrix with a scalar, only the non-zero elements in the diagonals can be multiplied and all other will remain as zeros.

$C=2\left[\begin{array}{ccc}4& 0& 0\\ 0& -3& 0\\ 0& 0& -1\end{array}\right]\phantom{\rule{0ex}{0ex}}C=\left[\begin{array}{ccc}8& 0& 0\\ 0& -6& 0\\ 0& 0& -2\end{array}\right]$

Hence, diagonal matrices are a subspace of ${\mathrm{\mathbb{R}}}^{3x3}$.

94% of StudySmarter users get better grades.

Sign up for free