Linear Review

Front

What is a nontrivial solution in the context of linear equations?

Back

A nontrivial solution is a solution to a linear equation that is not the zero vector. It indicates that there are infinitely many solutions to the equation.

Front

What does it mean for a matrix to have less than n pivot positions?

Back

It means that the matrix does not have full rank, which implies that the corresponding homogeneous equation has nontrivial solutions.

Front

Can the columns of a 3x5 matrix span R^5?

Back

No, a 3x5 matrix can have at most 3 linearly independent columns, so it cannot span R^5.

Front

What is the relationship between spanning and linear dependence?

Back

If the columns of a matrix span R^n, they are linearly independent. If they do not span R^n, they may be linearly dependent.

Front

What is a free variable in a system of linear equations?

Back

A free variable is a variable that can take on any value, leading to infinitely many solutions in a system of linear equations.

Front

What does it mean for columns of a matrix to be linearly independent?

Back

Columns of a matrix are linearly independent if no column can be expressed as a linear combination of the others.

Front

What is the significance of the rank of a matrix?

Back

The rank of a matrix indicates the maximum number of linearly independent column vectors in the matrix, which also reflects the dimension of the column space.

Front

What is the condition for a system of linear equations to have at least one solution?

Back

A system of linear equations has at least one solution if the rank of the coefficient matrix equals the rank of the augmented matrix.

Front

What does it mean for a matrix to be invertible?

Back

A matrix is invertible if it has full rank (n pivot positions for an nxn matrix), meaning the corresponding linear system has a unique solution.

Front

What is the geometric interpretation of linear independence?

Back

Geometrically, linear independence means that the vectors do not lie in the same line or plane; they span a higher-dimensional space.