(a) Show that the vectors ( 1 1 1 0 ),( 1 1 -1 0 ),(

(a) Show that the vectors ( 1 1 1 ...

Key Concepts

Systems of Linear Equations

Solving systems of linear equations is a fundamental tool used when expressing a vector as a linear combination of other vectors. In this context, determining the scalars that combine to form a given vector involves setting up and solving a set of simultaneous equations, which guarantees a unique solution if the vectors form a basis.

Basis

A basis for a vector space is a set of linearly independent vectors that spans the entire space. The concept of a basis is critical because it provides the minimal and most efficient set of directions needed to represent every element of the space uniquely as a combination of those vectors.

Linear Combination

A linear combination involves adding together scalar multiples of vectors. When expressing one vector as a linear combination of others, you are finding a set of scalar coefficients such that, when applied to the given vectors, their sum equals the target vector. This concept is essential for understanding coordinate representation in vector spaces.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be written as a linear combination of the others. This means that the only solution to the equation ax1 + bx2 + ... + nxn = 0 (with x1, x2, ..., xn being the vectors) is a1 = a2 = ... = an = 0. Establishing this property confirms that the vectors do not have redundant directions in the space.

Spanning Sets

A set of vectors spans a vector space if every vector in the space can be expressed as a linear combination of the set. In other words, the span of a set is the set of all possible linear combinations of its vectors. Demonstrating that a set of vectors spans the space shows that they cover the entire space, meaning any vector in the space can be reached using the set.

Transcript

00:01 Here we have given the vectors v1 is equal to 1, 2, 3, 4, v2, 0 ,0 minus 1, and v3 equal to 1 333.

00:16 Now to show the first part of this question is part a, is show that v1, v2, and v3 form a linearly dependent set in r4.

00:28 That means we have to show that these vectors are linearly dependent and part b is express each vector as a linear combination of the other two that means we need to express each vector v1 v2 and v3 in linear combination of other two vectors so the solution for this is given as solution now let us do part a first so part in part a we have v1 is equal to 1 2 3 4 v2 is equal to 0 1 0 minus 1 and v3 is equal to 1 3 3 3 now we need to show that these vector form a linearly dependent set.

01:28 That means any vector among the three vector any one vector among three vector can be written as the linear combination of other two vectors so let us take the vector v1 is equal to let v1 is equal to one two three four now this vector can be written as so we need to write the vector v1 a linear combination of vector v2 and b3 so what we need to do we need to find such scalar so that it can be retained in the linear combination of v2 and v3 so the scalar r so the scalar are minus one times of v2 plus one times of v3 which will give us minus 1 times of 0 1 0 minus 1 plus 1 times of 1 3 3 3 so on solving it we have so here it is 0 minus 1 0 plus 1 plus 1 3 3 so on adding these 2 vector we get 1 here it is 2 here it is 3 and here it is 4 which is nothing but our v1 so v1 can be written as so v1 can be written as so v1 can be written as minus 1 times of v2 plus 1 times of v3 which shows that it shows that v1 v2 and v3 and v3 are linearly linearly linearly dependent vectors in r -quib so this is the solution for our part a now moving towards part b we need to express each vector as a linear combination of other two vectors...

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