Math Problem Statement

Solve

Given vectors B1 = (1,1), B2 = (2,-1), and E = (1,0),(0,1), solve a series of problems including proving bases, expressing vectors in terms of bases, and computing coordinates.

Solution

I will solve the problems in the image step by step. Here's a structured solution:

1. Prove that B1B_1 and B2B_2 form bases in R2\mathbb{R}^2:

To determine if B1=(11)B_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} and B2=(2−1)B_2 = \begin{pmatrix} 2 \\ -1 \end{pmatrix} form a basis, we need to verify that they are linearly independent and span R2\mathbb{R}^2.

Linear Independence:

To check, solve the equation: aB1+bB2=0aB_1 + bB_2 = 0 a(11)+b(2−1)=(00)a \begin{pmatrix} 1 \\ 1 \end{pmatrix} + b \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} This expands to:

a + 2b = 0 \\ a - b = 0 \end{cases}$$ From $$a - b = 0$$, we have $$a = b$$. Substituting into $$a + 2b = 0$$, gives $$b + 2b = 0$$, hence $$b = 0$$ and $$a = 0$$. Thus, $$B_1$$ and $$B_2$$ are linearly independent. Since there are 2 vectors in $$\mathbb{R}^2$$, they form a basis. --- ### **2. Express the vectors $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$$ in terms of $$B_1, B_2$$:** Write $$\begin{pmatrix} 1 \\ 0 \end{pmatrix} = x_1B_1 + x_2B_2$$: $$x_1 \begin{pmatrix} 1 \\ 1 \end{pmatrix} + x_2 \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ Equations: $$x_1 + 2x_2 = 1 \quad (1)$$ $$x_1 - x_2 = 0 \quad (2)$$ From (2), $$x_1 = x_2$$. Substitute into (1): \[ x_2 + 2x_2 = 1 \implies 3x_2 = 1 \implies x

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Basis and Coordinates
Linear Independence
Change of Basis

Formulas

Linear combination: v = a1*v1 + a2*v2 + ...
Matrix representation of basis transformation
Determinant to check linear independence

Theorems

A set of vectors forms a basis if they are linearly independent and span the vector space.

Suitable Grade Level

Undergraduate (Linear Algebra course)

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