# Vectors, matrices, and baces

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Question 1

Let B={v1,...,vn} be a basis of a subspace V of Rnx1. Let x be the nonzero vector x=a1v1+...+anvn for scalars ai. Let C={x, v2,...,vn}.

a) Show that if a1 is not equal to 0 then C is also a basis.

b) Show that if a1 =0 then C is not a basis.

Question 2

Prove that if A is a real n by n matrix and if the expression xTAy for x,y elements of Rnx1 does indeed define an inner product <x| y>on Rnx1 then A must be symmetric and positive definite. Hint: consider <ei| ej>. Recall that a matrix A is positive definite if xTAx>0 for every nonzero vector x.

Question 3

Let V, W1, W2, and V0 be subspaces of Cnx1. Let B={u1, u2,...,un} be an orthonormal basis of V. Define V0┴ = {u | u ┴ v0 for all v0 that are elements of V0}. Observe (DO NOT PROVE) that

the intersection of W1 and W2, W1 + W2, and V0┴ are subspaces of V. By definition the sum

W1 + W2 is said to be direct if the intersection of W1 and W2 consists only of the zero vector.

a) Suppose that {u1,...,uk} (where k is less than or equal to n ) is an orthonormal basis of V0. Show that

V0┴ = span{uk+1,...,un}

And that

V0 + V0┴ = V.

b) Show that the sum V0 + V0┴ is a direct sum.

c) Show that if W1 + W2 is a direct sum then every vector w in W1 + W2 has a unique representation as w= w1+w2 for w1 in W1 and w2 in W2.

d) Conclude that every vector v in V has a unique representation as v = v0 + y with v0 in V0 and y in V0┴.

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##### Solution Summary

This provides examples of working with proofs regarding vectors and direct sums, symmetric and positive definite matrices, and bases.

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Question 1

Let B={v1,...,vn} be a basis of a subspace V of Rnx1. Let x be the nonzero vector x=a1v1+...+anvn for scalars ai. Let C={x, v2,...,vn}.

a) Show that if a1 is not equal to 0 then C is also a basis.

b) Show that if a1 =0 then C is not a basis.

Question 2

Prove that if A is a real n by n matrix and if the expression xTAy for x,y elements of Rnx1 does indeed define an inner product <x| y>on Rnx1 then A must be symmetric and positive definite. Hint: consider <ei| ej>. Recall that a matrix A is positive definite if xTAx>0 for every nonzero vector x.

Question 3

Let V, W1, W2, and V0 be subspaces of Cnx1. Let B={u1, u2,...,un} be an orthonormal basis of V. Define V0┴ = {u | u ┴ v0 for all v0 that are elements of V0}. Observe (DO NOT PROVE) that

the intersection of W1 and W2, W1 + W2, and V0┴ are subspaces of V. By definition the sum

W1 + W2 is said to be direct if the intersection of W1 and W2 consists only of the zero vector.

a) Suppose that {u1,...,uk} (where k is less than or equal to n ) is an orthonormal basis of V0. Show that

V0┴ = span{uk+1,...,un}

And that

V0 + V0┴ = V.

b) Show that the sum V0 + V0┴ is a direct sum.

c) Show that if W1 + W2 is a direct sum then every vector w in W1 + W2 has a unique representation as w= w1+w2 for w1 in W1 and w2 in W2.

d) Conclude that every vector v in V has a unique representation as v = v0 + y with v0 in V0 and y in V0┴.

Answer ...

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