Do questions 53 and 54Show if it is a subspace using these 3 steps:1. has to be equal to the 0 vector2. has to be closed under addition3. has to be closed under mulitplication **53. Proof** Let $ A $ be a fixed $ 2 \times 3 $ matrix. Prove that the set \[W = \left\{ \mathbf{x} \in \mathbb{R}^3 : A\mathbf{x} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right\}\]is not a subspace of $\mathbb{R}^3$.**54. Proof** Let $ A $ be a fixed $ m \times n $ matrix. Prove that the set\[W = \left\{ \mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = 0 \right\}\]is a subspace of $\mathbb{R}^n$.

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53. Proof Let A be a fixed 2 × 3 matrix. Prove that the set is not a subspace of R³. Answer W is a subspace of Rn. = = { x = R¹₁ AX = ["]} R³: Ax 54. Proof Let A be a fixed m × n matrix. Prove that the set W = {x € R" : Ax = 0}

53. Proof Let A be a fixed 2 × 3 matrix. Prove that the set is not a subspace of R³. Answer W is a subspace of Rn. = = { x = R¹₁ AX = ["]} R³: Ax 54. Proof Let A be a fixed m × n matrix. Prove that the set W = {x € R" : Ax = 0}

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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