Math Problem Statement

Soit f l'endomorphisme de ℝ³ dont la matrice dans la base canonique est donnée, avec une série de questions sur la sous-espace vectoriel, calcul de f(b) et f(c), détermination de base, et matrice de passage.

Solution

The problem involves a linear algebra exercise with the endomorphism ff in R3\mathbb{R}^3 described by a matrix AA in the canonical basis. Here's a breakdown and solution process for each question:

1. Show that E1E_1 is a vector subspace of R3\mathbb{R}^3:

E1={(x,y,z)∈R3:f(x,y,z)=(x,y,z)}E_1 = \{(x, y, z) \in \mathbb{R}^3 : f(x, y, z) = (x, y, z)\}.

This subspace consists of all vectors in R3\mathbb{R}^3 that are fixed under the transformation ff.

  • Start with A(xyz)=(xyz)A \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}.
  • Solve (A−I)(xyz)=(000)(A - I)\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, where II is the identity matrix.

The matrix A−IA - I is:

0 & 2 & -2 \\ 2 & 0 & -2 \\ 2 & 2 & -4 \end{pmatrix}.$$ Find the kernel (null space) of $$A - I$$ to determine $$E_1$$. This will give a basis $$a$$. --- ### 2. **Calculate $$f(b)$$ and $$f(c)$$ for $$b = (0, 1, 1)$$ and $$c = (1, 1, 2)$$:** Compute $$f(b) = A \cdot b$$ and $$f(c) = A \cdot c$$, where: $$A = \begin{pmatrix} 1 & 2 & -2 \\ 2 & 1 & -2 \\ 2 & 2 & -3 \end{pmatrix}.$$ Perform the matrix multiplication for each vector. --- ### 3. **Show that $$B' = (a, b, c)$$ is a basis of $$\mathbb{R}^3$$:** Prove that the vectors $$a, b, c$$ are linearly independent by forming the matrix $$M = \begin{pmatrix} a & b & c \end{pmatrix}$$ and checking if $$\det(M) \neq 0$$. --- ### 4. **Find the transition matrix $$P$$ from $$B$$ to $$B'$$:** The transition matrix $$P$$ satisfies: $$P = \begin{bmatrix} | & | & | \\ a & b & c \\ | & | & | \end{bmatrix}.$$ Express $$a, b, c$$ in terms of the canonical basis vectors. --- ### 5. **Calculate $$P^{-1}$$:** Invert the matrix $$P$$ to find $$P^{-1}$$. --- If you'd like, I can work through these calculations step by step, including solving the null space for $$E_1$$, performing the matrix multiplications, and computing determinants or inverses. Let me know! ### Expanding on the Concepts: 1. How do we confirm a set is a vector subspace of $$\mathbb{R}^3$$? 2. What methods can we use to compute the null space of a matrix efficiently? 3. What properties determine linear independence of vectors in $$\mathbb{R}^3$$? 4. How is a transition matrix between bases defined in $$\mathbb{R}^3$$? 5. What are the key steps for inverting a 3x3 matrix manually? **Tip:** When working with subspaces, always verify closure under addition and scalar multiplication explicitly.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Matrix Transformation
Basis and Transition Matrices

Formulas

f(x, y, z) = (x, y, z)
Matrix-vector multiplication
Transition matrix formula
Inverse of a 3x3 matrix

Theorems

Properties of Vector Subspaces
Criteria for Linear Independence
Basis Change and Transition Matrix

Suitable Grade Level

Undergraduate

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