Math Problem Statement

just the solutions no explinations

Problems 43 to 47 in linear algebra, involving systems of linear equations, vector spaces, and matrix operations.

Solution

Here are the solutions for the given problems:

43.

  • Correct answer: a.

44.

  • Answers:
    • a. 1
    • b. 3
    • c. 4

45.

  • k=2k = 2

46.

  • Correct answer: b.

Let me know if you'd like further details or have additional questions!

Here are five related questions for further exploration:

  1. What does it mean for a set of vectors to be linearly independent?
  2. How can you determine the rank of a matrix?
  3. What are the properties of a vector space in linear algebra?
  4. Can you describe the difference between span and basis in vector spaces?
  5. What is the significance of eigenvalues in solving systems of linear equations?

Tip: When solving linear systems, reducing the matrix to row echelon form simplifies the process of finding solutions.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Vector Spaces
Matrix Operations

Formulas

Linear independence condition: det(A) ≠ 0
Solving system of equations: AX = B
Linear combinations of vectors
Inverse matrix formula: (AB)^-1 = B^-1A^-1

Theorems

Linear Independence Theorem
Rank-Nullity Theorem
Properties of Inverses of Matrices

Suitable Grade Level

Undergraduate level - Linear Algebra