Linear Algebra Concepts Flashcards

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Questions and Answers

What must you know about the pivot columns in an augmented matrix to determine that the linear system is consistent and has a unique solution?

Every column in the augmented matrix except the rightmost column is a pivot column, and the rightmost column is not a pivot column.

Any list of five real numbers is a vector in R5.

True

The vector u results when a vector u - v is added to the vector v.

True

The weights c1,...,cp in a linear combination c1v1 + ... + cpvp cannot all be zero.

False Signup and view all the answers

When u and v are nonzero vectors, Span {u, v} contains the line through u and the origin.

True Signup and view all the answers

Asking whether the linear system corresponding to an augmented matrix [a1, a2, a3, b] has a solution amounts to asking whether b is in Span {a1, a2, a3}.

True Signup and view all the answers

What is the number of vectors in the set {a1, a2, a3}?

There are only three vectors in the set. Signup and view all the answers

Can each vector in R4 be written as a linear combination of the columns of the matrix A?

No. Signup and view all the answers

Can a set of three vectors in R4 span all of R4?

No. Signup and view all the answers

Why must the columns of matrix A span R3 if Ax = b has a unique solution?

If Ax = b has a unique solution, then the associated system does not have any free variables, implying every column of A is a pivot column. Signup and view all the answers

What fact allows you to conclude that the system Ax = w is consistent?

Using theorem 5(a), Ax1 + Ax2 = A(x1 + x2), shows x = x1 + x2 is a solution of w = Ax. Signup and view all the answers

What fact allows you to conclude that the system Ax = 4z is consistent?

4z = A(4y) shows there is a solution since Ax = z has a known solution. Signup and view all the answers

Explain why the solution to Ax = b is unique precisely when Ax = 0 has only the trivial solution.

There are no free variables in the system of equations if Ax = 0 has only the trivial solution. Signup and view all the answers

Find one nontrivial solution of Ax = 0 by inspection for A = [ 2 x 3 ].

x = [3, -1] Signup and view all the answers

Explain why A(u + v) must be the zero vector given Au = 0 and Av = 0.

A(u + v) = Au + Av = 0 + 0 = 0. Signup and view all the answers

Study Notes

Linear Systems and Augmented Matrices

A linear system is consistent with a unique solution if every column in the augmented matrix (excluding the rightmost column) is a pivot column.
In an augmented matrix, the rightmost column should not be a pivot column for consistency with a unique solution.

Vectors and Linear Combinations

Any list of five real numbers qualifies as a vector in R⁵.
Adding the vector u-v to vector v results in vector u.
The weights in a linear combination can include all zeros, thus allowing for the possibility of a solution that is not constrained by non-zero weights.
The span of nonzero vectors u and v includes the line connecting u to the origin.

Span and Solutions

Asking if the linear system represented by an augmented matrix has a solution relates to whether the vector b is included in the Span of vectors a₁, a₂, and a₃.
A set of three vectors in R⁴ cannot span all of R⁴ since a matrix made up of these vectors lacks enough pivot positions for every dimension.
A unique solution for Ax = b indicates that all columns of A are pivot columns, thus spanning R³.

Theorem Applications in Consistency

The linear combination of two vector solutions results in a valid solution for the system Ax = w, confirming consistency.
For the system Ax = 4z to be consistent, it is derived from a similar previous solution, applying scalar multiplication.

Uniqueness of Solutions

A solution to Ax = b is unique if Ax = 0 only has the trivial solution, which occurs when every column is a pivot column.
Nontrivial solutions to Ax = 0 can be identified through relationships between the matrix's columns and independent choices for vector components.

Properties of Linear Transformations

If vectors u and v satisfy Au = 0 and Av = 0, their sum A(u + v) remains a zero vector, leading to the conclusion that any linear combination A(cu + dv) also results in zero.

These notes should provide clear insights into concepts surrounding linear systems, augmented matrices, and vector relationships, as well as implications regarding solution uniqueness and consistency.

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Test your knowledge on key concepts in linear algebra through these flashcards. Each card includes essential definitions and true/false questions related to vectors and augmented matrices. Challenge yourself to understand the foundations of linear system solutions!