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Mark each statement as true or False. Justify your answer. If true, cite a theorem or fact from class. If false, explain why by giving the correct answer or a counter example. (a) Every line of the form y = mx + b is a subspace of R². (b) A is a 4 x 4 matrix, and its reduced matrix is the identity matrix, then its determinant is 4. (c) Every linearly independent set of vectors in a vector space V, can be extended to a basis for V. (d) If T : R³ → R² is a linear transformation, a basis of the null space can have at most two vectors. (e) Column space(A) = Column space(B), where B is the reduced form of A.

Mark each statement as true or False. Justify your answer. If true, cite a theorem or fact from class. If false, explain why by giving the correct answer or a counter example. (a) Every line of the form y = mx + b is a subspace of R². (b) A is a 4 x 4 matrix, and its reduced matrix is the identity matrix, then its determinant is 4. (c) Every linearly independent set of vectors in a vector space V, can be extended to a basis for V. (d) If T : R³ → R² is a linear transformation, a basis of the null space can have at most two vectors. (e) Column space(A) = Column space(B), where B is the reduced form of A.

Advanced Engineering Mathematics
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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Mark each statement as true or False. Justify your answer. If true, cite a theorem or fact from class. If false, explain why by giving the correct answer or a counter example. (a) Every line of the form y = mx + b is a subspace of R². (b) A is a 4 x 4 matrix, and its reduced matrix is the identity matrix, then its determinant is 4. (c) Every linearly independent set of vectors in a vector space V, can be extended to a basis for V. (d) If T : R³ → R² is a linear transformation, a basis of the null space can have at most two vectors. (e) Column space(A) = Column space(B), where B is the reduced form of A.
Transcribed Image Text:Mark each statement as true or False. Justify your answer. If true, cite a theorem or fact from class. If false, explain why by giving the correct answer or a counter example. (a) Every line of the form y = mx + b is a subspace of R². (b) A is a 4 x 4 matrix, and its reduced matrix is the identity matrix, then its determinant is 4. (c) Every linearly independent set of vectors in a vector space V, can be extended to a basis for V. (d) If T : R³ → R² is a linear transformation, a basis of the null space can have at most two vectors. (e) Column space(A) = Column space(B), where B is the reduced form of A.
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