Let V, W be vector spaces and let U be a subspace of W.(a) Is the set {f ∈ L (V, W) : U ⊂ im f} a subspace of L (V, W)?(b) What about the set S(U) = {f ∈ L (V, W) : im f ⊂ U}?(c) Assuming that V and W are finite-dimensional, find dim S(U) in terms of dim V, dim U, and dim W

Let V, W be vector spaces and let U be a subspace of W.(a) Is the set {f ∈ L (V, W) : U ⊂ im f} a subspace of L (V, W)?(b) What about the set S(U) = {f ∈ L (V, W) : im f ⊂ U}?(c) Assuming that V and W are finite-dimensional, find dim S(U) in terms of dim V, dim U, and dim W

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 26EQ

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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Let V, W be vector spaces and let U be a subspace of W.
(a) Is the set {f ∈ L (V, W) : U ⊂ im f} a subspace of L (V, W)?
(b) What about the set S(U) = {f ∈ L (V, W) : im f ⊂ U}?
(c) Assuming that V and W are finite-dimensional, find dim S(U) in terms of dim V, dim U, and dim W

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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