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Found in: Page 176

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Which of the subsets V of ${{\mathbf{ℝ}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$ given in Exercise 6 through are subspaces of ${{\mathbf{ℝ}}}^{\mathbf{3}\mathbf{x}\mathbf{3}}$. The upper triangular 3x3 matrices.

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

See the step by step solution

## Step 1: Definition of subspace.

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.

## Step 2: Verification whether the subset is closed under addition and scalar multiplication.

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 ${\mathrm{ℝ}}^{3x3}$.