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Are the following statements true or false? + 1. The set (0) forms a basis for the zero subspace. + 2. If Si is of dimension 3 and is a subspace of R*, then there can not exist a subspaceS2 of R* such that S1 CS2C R* with S1 # S2 and S2# R*. A 3. Let m > n. Then U = {u1, u). . Um) in R" can form a basis for R" if the correct m -n vectors are removed from U. + 4. Let m < n. Then U = {u1, u2... , um} in R" can form a basis for R" if the correct n – m vectors are added to U. + 5. R" has exactly one subspace of dimension m for each of m = 0, 1,2, ..., n.

Are the following statements true or false? + 1. The set (0) forms a basis for the zero subspace. + 2. If Si is of dimension 3 and is a subspace of R*, then there can not exist a subspaceS2 of R* such that S1 CS2C R* with S1 # S2 and S2# R*. A 3. Let m > n. Then U = {u1, u). . Um) in R" can form a basis for R" if the correct m -n vectors are removed from U. + 4. Let m < n. Then U = {u1, u2... , um} in R" can form a basis for R" if the correct n – m vectors are added to U. + 5. R" has exactly one subspace of dimension m for each of m = 0, 1,2, ..., n.

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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Are the following statements true or false? + 1. The set (0) forms a basis for the zero subspace. + 2. If Si is of dimension 3 and is a subspace of R*, then there can not exist a subspaceS2 of Rª such that Si C S2 C Rª with S1 # S2 and S2# R*. 3. Let m > n. Then U = {u], un. ...,Um) in R" can form a basis for R" if the correct m –n vectors are removed from U. + 4. Let m < n. Then U = {u],u2 ... , Um in R" can form a basis for R" if the correct n – m vectors are added to U. + 5. R" has exactly one subspace of dimension m for each of m = 0, 1,2, ... , n .
Transcribed Image Text:Are the following statements true or false? + 1. The set (0) forms a basis for the zero subspace. + 2. If Si is of dimension 3 and is a subspace of R*, then there can not exist a subspaceS2 of Rª such that Si C S2 C Rª with S1 # S2 and S2# R*. 3. Let m > n. Then U = {u], un. ...,Um) in R" can form a basis for R" if the correct m –n vectors are removed from U. + 4. Let m < n. Then U = {u],u2 ... , Um in R" can form a basis for R" if the correct n – m vectors are added to U. + 5. R" has exactly one subspace of dimension m for each of m = 0, 1,2, ... , n .
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