Linear Algebra Exam Practice Part 1: Linear System of Equations Problem Solutions

Get ready for your linear algebra exam with this step-by-step walkthrough of Problems 1–13 from our comprehensive review!
1. Determine if a system has no solution, one solution, or infinitely many solutions, and solve if possible.
2. Identify whether matrices are in echelon form, reduced echelon form, or neither.
3. Find values of h and k that result in systems with no solution, a unique solution, or infinitely many solutions.
4. Use Gauss-Jordan elimination to find the general solution of a system of equations.
5. Solve a system of equations using an augmented matrix and express the general solution in parametric form.
6. Provide a geometric interpretation of the span of three vectors.
7. Determine the value of h that ensures a vector is in the span of two other vectors.
8. Write a system of equations as a matrix equation and a vector equation.
9. Identify a set of vectors Ax=b that has a solution and show why Ax=b does not always have a solution.
10. True/False Statements on Matrix Properties: Evaluate statements about pivots, spans, and linear combinations.
11. Solve matrix equations Ax=0 and Ax=b, providing solutions in parametric vector form with geometric interpretation.
12. Determine values of h that make a set of vectors linearly dependent.
13. Analyze sets of vectors to determine if they are linearly independent or dependent.
This step-by-step problem walkthrough is perfect for preparing for your linear algebra midterm or final exam!

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*Timestamps*
00:00:00 Introduction
00:00:23 Problem 1
00:03:46 Problem 2
00:07:49 Problem 3
00:12:09 Problem 4
00:20:46 Problem 5
00:27:01 Problem 6
00:33:01 Problem 7
00:38:01 Problem 8
00:40:22 Problem 9
00:43:48 Problem 10
00:49:30 Problem 11
00:58:41 Problem 12
01:04:02 Problem 13

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[abdullahalotaibi]

In Question 10b, If every coloum has a pivot then technically every row has a pivot so it should be true right. Unless, the question explicitly says only when A has a pivot in every column which is not the case. Please correct me If I am wrong, Thank you.

[JanVriesman]

For question 10a, if matrix A = [1 2, 0 0] and b = [3, 0], A doesn’t have a pivot position in every row but the equation Ax = b has a solution, right?