Linear Algebra Flashcards
What is a pivot? What is a pivot column?
A pivot is the first non-zero entry of each row if the a matrix is in row-echelon form.
A pivot column is just a column that contains a pivot
What is a “free” variable vs a pivot variable (also known as basic variable)?
A pivot/basic variable is a variable associated with a pivot column. A free variable is a variable a free column, a column that doesn’t have a pivot
You can think of free variables as being independent and non-free variables as being dependent
What is homogenous system in linear algebra?
What solutions do/can homogenous systems have?
A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. When a row operation is applied to a homogeneous system, the new system is still homogeneous.
Ax = a zero vector
Homogenous systems always have the trivial solution, a zero vector. They can also have non-trivial solutions if matrix A has a free variable/has a column with no pivot
What is the paremetric form/parametric equation of a solution set?
x = s1v1 + s2v2 + ··· + smvm.
Where v’s are vectors and s’s are the weights/coefficients and x is a vector
What does it mean to say that a set of vectors are linearly independent?
If a set of vectors are linearly independent, that means that the linear combination of the vectors equals a zero vector only with the trivial solution. If the linear combination equals a zero vector with any other solution this means that the vectors are linearly dependent.
What does “span” refer to in linear algebra?
Span refers to the set of all linear combinations of a set of vectors. So for a set of vectors, the span refers to the set of solutions
Span refers to the set of every linear combination of a set of vectors.
https://mikebeneschan.medium.com/how-to-understand-span-linear-algebra-cf3baa12edda
How does the dependence/independence of vectors impact the span of the linear combination of those vectors?
If vectors are dependent, the span is the same as if we remove one of the vectors. If vectors are independent, the span changes if you remove a vector.
How can you tell if two vectors are dependent?
Two vectors are dependent if they are collinear, meaning that one vector is a scalar multiple of the other vector. They are also dependent if one or both of the vectors is the zero vector.
How does the span of a set of vectors relate to dependence/independence of the vectors?
If a set of vectors are dependent, then at least 1 of the vectors is redundant, meaning it can be removed without impacting the span.
If a set of vectors are independent then each vector adds to the span of the set of vectors.
If b is not a linear combination of the columns of A, then can Ax = b be consistent?
No, if be isn’t a linear combination then that system does not have a solution.
Matrix A has linearly independent columns and Ax = 0 has infinite solutions.
Possible or impossible?
Impossible; if the columns are independent then every column is pivotal and thus only the trivial solution solves Ax = 0
What are the standard vectors for linear transformations?
Vectors that when multiplied by a matrix give you only a single column from the matrix. So they have a single 1 and the rest zeroes
What is a one-to-one transformation?
A matrix Transformation T is one-to-one if Matrix A has a pivot in every _____.
Definition(One-to-one transformations)
A transformation T:Rn→Rm is one-to-one if, for every vector b in Rm, the equation T(x)=b has at most one solution x in Rn. In other words, Ax = b has at most 1 solution. So for the zero vector b there is only the trivial solution.
One-to-one is a uniqueness property, it does not assert existence for all vectors b.
One-to-one, meaning there isn’t more than 1 transformation that is possible for each vector.
A matrix Transformation T is one-to-one if Matrix A has a pivot in every column.
What is an onto transformation?
A matrix transformation T is onto if Matrix A has a pivot in every ______.
Definition(Onto transformations)
A transformation T:Rn→Rm is onto if, for every vector b in Rm, the equation T(x)=b has at least one solution x in Rn. So for every vector b there is a vector x such that Ax = b has a solution (Ax = b is always consistent)
Onto is an existence property
A matrix transformation T is onto if and only if Matrix A has a pivot in every row.
True or false.
If the only solution to Ax = 0 is x = 0, then T(x) must a one to one transformation
True; if the only solution is the trivial solution then every column of A must be pivotal
