LINEAR ALGEBRA EXAM 1 CONCEPTS
AND PROBLEM-SOLVING OUTLINES-
QUESTIONS WITH CORRECT ANSWERS
Matrix Equation - Answer-Ax = b
How to solve a matrix equation's set of equations? - Answer-Row reduce
Given augmented matrix, how to write down general solution of system in parametric
form? - Answer-Row reduce, find free variables, describe other variables in terms of
free variables (general solution, and look at columns not rows), use variable name as
weight and then coefficients as corresponding vector, add right side of the equation
variables as standalone vector with no coefficient variable
What does it mean if a system is consistent? Always consistent? - Answer-It has one
unique (nontrivial) solution or infinitely many solutions
If there's a pivot in every column, what does that mean? - Answer-The vectors are
linearly independent.
If there's a pivot in every row, what does that mean? - Answer-The vectors span their
given dimension.
How do I write E1 as a linear combination of two given vectors? - Answer-Form
augmented matrix with vector 1, vector 2, and E1 (which is always 1, 0); then row
reduce to find if system is consistent and if so, what the solution is. Then use c1v1 +
c2v2 general formula where the c coefficients are the ones you just found, and the v
variables are the second T given to you after the first T vector. That vector answer will
be a part of the matrix for T. If looking for another vector within same transformation,
can just multiply by the matrix (A"newx" = b)
If a matrix equation has a nontrivial solution, what is it not? - Answer-It is not one-to-
one.
A homogeneous system has only the trivial solution when: - Answer-Every column has
a pivot.
A homogeneous system cannot always span the real dimension if: - Answer-There is
not a pivot in every row.
Number of pivots corresponds with: - Answer-lowest number between m or n
AND PROBLEM-SOLVING OUTLINES-
QUESTIONS WITH CORRECT ANSWERS
Matrix Equation - Answer-Ax = b
How to solve a matrix equation's set of equations? - Answer-Row reduce
Given augmented matrix, how to write down general solution of system in parametric
form? - Answer-Row reduce, find free variables, describe other variables in terms of
free variables (general solution, and look at columns not rows), use variable name as
weight and then coefficients as corresponding vector, add right side of the equation
variables as standalone vector with no coefficient variable
What does it mean if a system is consistent? Always consistent? - Answer-It has one
unique (nontrivial) solution or infinitely many solutions
If there's a pivot in every column, what does that mean? - Answer-The vectors are
linearly independent.
If there's a pivot in every row, what does that mean? - Answer-The vectors span their
given dimension.
How do I write E1 as a linear combination of two given vectors? - Answer-Form
augmented matrix with vector 1, vector 2, and E1 (which is always 1, 0); then row
reduce to find if system is consistent and if so, what the solution is. Then use c1v1 +
c2v2 general formula where the c coefficients are the ones you just found, and the v
variables are the second T given to you after the first T vector. That vector answer will
be a part of the matrix for T. If looking for another vector within same transformation,
can just multiply by the matrix (A"newx" = b)
If a matrix equation has a nontrivial solution, what is it not? - Answer-It is not one-to-
one.
A homogeneous system has only the trivial solution when: - Answer-Every column has
a pivot.
A homogeneous system cannot always span the real dimension if: - Answer-There is
not a pivot in every row.
Number of pivots corresponds with: - Answer-lowest number between m or n
