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Linear Algebra and its Applications

Found in: Page 191

Book edition 5th

Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald

Pages 483 pages

ISBN 978-03219822384

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Short Answer

Let H be an n-dimensional subspace of an n-dimensional vector space V. Explain why \(H = V\).

H is a subspace of V, and B is also in V. As B is a linearly independent set in V, B must also be a basis for V.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the dimension of H and V

The bases of H and V have exactly n vectors because H is an n-dimensional subspace of V.

For \(n = 0\),

\(H = V = \left\{ 0 \right\}\).

Step 2: Check for \(H = V\)

For subspace H is n-dimensional subspace, there is a B for H. B must have n elements and be linearly independent.

Since H is a subspace of V and B is also in V, B is a linearly independent set in V. So, B must also be a basis for V. Hence, H and V are the same.

Most popular questions for Math Textbooks

[M] Repeat Exercise 35 for a random integer-valued matrix whose rank is at most 4. One way to make is to create a random integ\(6 \times 7\)er-valued \(6 \times 4\) matrix \(J\) and a random integer-valued \(4 \times 7\) matrix \(K\), and set \(A = JK\). (See supplementary Exercise 12 at the end of the chapter; and see the study guide for the matrix-generating program.)

(M) Show that \(\left\{ {t,sin\,t,cos\,{\bf{2}}t,sin\,t\,cos\,t} \right\}\) is a linearly independent set of functions defined on \(\mathbb{R}\). Start by assuming that

\({c_{\bf{1}}} \cdot t + {c_{\bf{2}}} \cdot sin\,t + {c_{\bf{3}}} \cdot cos\,{\bf{2}}t + {c_{\bf{4}}} \cdot sin\,t\,cos\,t = {\bf{0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\bf{5}} \right)\)

Equation (5) must hold for all real t, so choose several specific values of t (say, \(t = {\bf{0}},\,.{\bf{1}},\,.{\bf{2}}\)) until you get a system of enough equations to determine that the \({c_j}\) must be zero.

(M) Show that is a linearly independent set of functions defined on . Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)

If A is a \({\bf{6}} \times {\bf{4}}\) matrix, what is the smallest possible dimension of Null A?

Let \({M_{2 \times 2}}\) be the vector space of all \(2 \times 2\) matrices, and define \(T:{M_{2 \times 2}} \to {M_{2 \times 2}}\) by \(T\left( A \right) = A + {A^T}\), where \(A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\).

Show that \(T\)is a linear transformation.
Let \(B\) be any element of \({M_{2 \times 2}}\) such that \({B^T} = B\). Find an \(A\) in \({M_{2 \times 2}}\) such that \(T\left( A \right) = B\).
Show that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\) with the property that \({B^T} = B\).
Describe the kernel of \(T\).

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