True/False. Answer in the blanks next to the statements. a) A basis is always a spanning set. b) Since the dimension ofR? is 2, then the dimension of any subspace of R must also be 2. c) If a set of three vectors is linearly dependent, then one vector can be written as a linear combination of the other two. d) If S is a set of linearly dependent vectors in a vector space V, then span(S) is a subspace of V. e) If a matrix A is not a square matrix, the dimension of the row space of A and the dimension of the column space of A are different. f) If S {v1, v2, v3, v4} is a basis for a vector space V, then other bases for V can have fewer than four vectors, but not more.

True/False. Answer in the blanks next to the statements. a) A basis is always a spanning set. b) Since the dimension ofR? is 2, then the dimension of any subspace of R must also be 2. c) If a set of three vectors is linearly dependent, then one vector can be written as a linear combination of the other two. d) If S is a set of linearly dependent vectors in a vector space V, then span(S) is a subspace of V. e) If a matrix A is not a square matrix, the dimension of the row space of A and the dimension of the column space of A are different. f) If S {v1, v2, v3, v4} is a basis for a vector space V, then other bases for V can have fewer than four vectors, but not more.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

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.

Question

True/False. Answer in the blanks next to the statements.
a) A basis is always a spanning set.
b) Since the dimension ofR is 2, then the dimension of any
subspace of R must also be 2.
c) If a set of three vectors is linearly dependent, then one vector
can be written as a linear combination of the other two.
d) If S is a set of linearly dependent vectors in a vector space V,
then span(S) is a subspace of V.
e) If a matrix A is not a square matrix, the dimension of the row
space of A and the dimension of the column space of A are
different.
f) If S = {, v2, v3, v} is a basis for a vector space V, then other
bases for V can have fewer than four vectors, but not more.

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