Find the largest possible number of independent vectors...

Find the largest possible number of independent vectors among $$ v_{1}=\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ 0 \end{array}\right] \quad v_{2}=\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ 0 \end{array}\right] \quad v_{3}=\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -1 \end{array}\right] v_{4}=\left[\begin{array}{r} 0 \\ 1 \\ -1 \\ 0 \end{array}\right] \quad v_{5}=\left[\begin{array}{r} 0 \\ 1 \\ 0 \\ -1 \end{array}\right] v_{6}=\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ -1 \end{array}\right] $$ This number is the of the space spanned by the v's.

Find the largest possible number of independent vectors among
$$
v_{1}=\left[\begin{array}{r}
1 \\
-1 \\
0 \\
0
\end{array}\right] \quad v_{2}=\left[\begin{array}{r}
1 \\
0 \\
-1 \\
0
\end{array}\right] \quad v_{3}=\left[\begin{array}{r}
1 \\
0 \\
0 \\
-1
\end{array}\right] v_{4}=\left[\begin{array}{r}
0 \\
1 \\
-1 \\
0
\end{array}\right] \quad v_{5}=\left[\begin{array}{r}
0 \\
1 \\
0 \\
-1
\end{array}\right] v_{6}=\left[\begin{array}{r}
0 \\
0 \\
1 \\
-1
\end{array}\right]
$$
This number is the of the space spanned by the v's.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Linear Independence

This concept refers to a set of vectors where no vector in the set can be written as a linear combination of the others. In other words, the only solution to the equation involving a linear combination of these vectors being equal to the zero vector is when all the coefficients are zero. This is essential when determining the maximum number of independent vectors in any set.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. It represents a subspace of the vector space, and understanding the span helps in determining how much of the space is covered by the set of vectors. Knowing the span is key to analyzing problems related to basis and dimension.

Dimension of a Vector Space

The dimension is defined as the maximum number of linearly independent vectors in a vector space. It is a fundamental concept in linear algebra that quantifies the number of directions in the space and is directly related to the idea of a basis for that space.

Rank of a Matrix

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In the context of vector sets, it corresponds to the dimension of the space that the vectors span. It provides a systematic way to determine the number of independent vectors among a given set.

Transcript

Please give Ace some feedback