Vector Space and Subspace of 2x2 Matrices: Basis and Dimension

Math Problem Statement

Sia X lo spazio vettoriale delle matrici reali 2x2 e si consideri il seguente sottoinsieme di X: Y := { A ∈ X : A(1 -1)^T = (0 0)^T }. 1. Si provi che Y è un sottospazio di X. 2. Determinare una base di Y e la dimensione di Y. 3. Sia Z il sottospazio di X delle matrici A tale che im(A) ⊂ ⟨(1 2)^T⟩. Determinare una base e la dimensione di Y ∩ Z.

Solution

Let's break down the exercise from the image into manageable steps:

Problem Summary:

Given the vector space $X$ of real $2 \times 2$ matrices, we are dealing

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Subspaces
Matrices

Formulas

Matrix multiplication A(1 -1)^T = (0 0)^T
Image of a matrix im(A) ⊂ ⟨(1 2)^T⟩

Theorems

Subspace Theorem
Basis and Dimension Theorem

Suitable Grade Level

Undergraduate Level

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