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Q8E

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 **** are subspaces of ${{\mathbf{\mathbb{R}}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$****. The upper triangular 3x3**** matrices.**

The upper triangular 3x3 matrices subset V of${\mathbf{\mathbb{R}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$ is a subspace of ${\mathbf{\mathbb{R}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$.

**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 upper triangular matrix namely,

$A=\left[\begin{array}{ccc}1& 4& -1\\ 0& 2& 5\\ 0& 0& 3\end{array}\right]\phantom{\rule{0ex}{0ex}}B=\left[\begin{array}{ccc}4& 1& 3\\ 0& -3& 2\\ 0& 0& -1\end{array}\right]$

Find A+B.

$A+B=\left[\begin{array}{ccc}1& 4& -1\\ 0& 2& 5\\ 0& 0& 3\end{array}\right]+\left[\begin{array}{ccc}4& 1& 3\\ 0& -3& 2\\ 0& 0& -1\end{array}\right]\phantom{\rule{0ex}{0ex}}A+B=\left[\begin{array}{ccc}5& 5& 2\\ 0& -1& 7\\ 0& 0& 2\end{array}\right]$

Thus, the sum of two upper triangular matrices is also an upper triangular matrix, because all other entries will be zero and only the entries in the upper half can be added and remain as non-zero elements.

Similarly, when multiplying an upper triangular matrix with a scalar, only the non-zero elements in the upper half of the diagonal can be multiplied and all other will remain as zeros.

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

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

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