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Let • Find rank(A) • Find basis for row(A) • Find basis for col(A) • Find basis for null(A) • Find basis for null(A¹) 2-1 1 -6 8 -2-4 3 -2 -7 8 10 3 -10 4 -5 -7 0 4 1 • Check that row(A) I null(A) by taking a vector from each subspace and seeing if the dot product is zero. • Check that col(A) I null(A¹) by taking a vector from each subspace and seeing if the dot product is zero.

Let • Find rank(A) • Find basis for row(A) • Find basis for col(A) • Find basis for null(A) • Find basis for null(A¹) 2-1 1 -6 8 -2-4 3 -2 -7 8 10 3 -10 4 -5 -7 0 4 1 • Check that row(A) I null(A) by taking a vector from each subspace and seeing if the dot product is zero. • Check that col(A) I null(A¹) by taking a vector from each subspace and seeing if the dot product is zero.

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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2. A linearly independent set (v₁, ... , Vk) in R" can be extended to a basis for R". One way to do this is to create A = [v₁.Vk е₁еn], with e₁, ..., e, the columns of the identity matrix. The columns containing the leading 1's in the RREF of A correspond to the columns of A needed to create a basis for R". Use this method to extend the following vectors to a basis for R³: V₁ = -7 -5 V2 = 9 4 1 6 V3 = 7 -8
Transcribed Image Text:2. A linearly independent set (v₁, ... , Vk) in R" can be extended to a basis for R". One way to do this is to create A = [v₁.Vk е₁еn], with e₁, ..., e, the columns of the identity matrix. The columns containing the leading 1's in the RREF of A correspond to the columns of A needed to create a basis for R". Use this method to extend the following vectors to a basis for R³: V₁ = -7 -5 V2 = 9 4 1 6 V3 = 7 -8
1. Let • Find rank(A) • Find basis for row(A) • Find basis for col(A) • Find basis for null(A) A = 2-1 1 -6 1 -2 -4 -7 8 10 4 -5 -7 0 8 3 -2 3 -10 • Find basis for null(A¹) • Check that row(A) 1 null(A) by taking a vector from each subspace and seeing if the dot product is zero. • Check that col(A) I null(A¹) by taking a vector from each subspace and seeing if the dot product is zero.
Transcribed Image Text:1. Let • Find rank(A) • Find basis for row(A) • Find basis for col(A) • Find basis for null(A) A = 2-1 1 -6 1 -2 -4 -7 8 10 4 -5 -7 0 8 3 -2 3 -10 • Find basis for null(A¹) • Check that row(A) 1 null(A) by taking a vector from each subspace and seeing if the dot product is zero. • Check that col(A) I null(A¹) by taking a vector from each subspace and seeing if the dot product is zero.
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