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In Exercises 19 and 20, V is a vector space. Mark each statement True or False. Justify each answer.

19.

a. The number of pivot columns of a matrix equals the dimension of its column space.

b. A plane in R3 is a two-dimensional subspace of R3.

c. The dimension of the vector space P4 is 4.

d. If dimโกV=n and S is a linearly independent set in V, then S is a basis for V.

e. If a set {v1,...,vp} spans a finite-dimensional vector space V and if T is a set of more than p vectors in V, then T is linearly dependent.

Short Answer

Expert verified
  1. The given statement is true.
  2. The given statement is false.
  3. The given statement is false.
  4. The given statement is false.
  5. The given statement is true.

Step by step solution

01

Determine whether the given statement is true or false

a)

Thedimension of NulA is thenumber of free variablesin the equation Ax=0, and the dimension of ColAis thenumber of pivot columnsin A.

Thus, the given statement (a) is true.

02

Determine whether the given statement is true or false

b)

A plane in R3 is athree-dimensional subspace of R3.

Thus, the given statement (b) is false.

03

Determine whether the given statement is true or false

c)

Thestandard basisfor Rn contains n vectors; so dimโกRn=n. The standard polynomial basis {1,t,t2} shows that P2=3. In general, dimโกPn=n+1.

Thus, the given statement (c) is false.

04

Determine whether the given statement is true or false

d)

Theorem 12states that let V be a p-dimensional vector space; pโ‰ฅ1, then any linearly independent set of exactly p elements in Vis automatically a basis for V. Any set of exactly p elements that span Vis automatically a basis for V.

The set S must have n elements.

Thus, the given statement (d) is false.

05

Determine whether the given statement is true or false

e)

Theorem 9states that if a vector space V has abasis B={b1,...,bn}, then any set in Vcontaining more than n vectors must belinearly dependent.

Thus, the given statement (e) is true.

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