Do the following sets of vectors span \( \mathbb{R}^{3} \) ? Select an Answer 1. \[ \left[\begin{array}{c} 2 \\ -1 \\ -2 \end{array}\right],\left[\begin{array}{c} 2 \\ -1 \\ -3 \end{array}\right],\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] \] Select an Answer 2. \[ \left[\begin{array}{c} 1 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{c} 5 \\ -7 \\ 5 \end{array}\right],\left[\begin{array}{c} 6 \\ -3 \\ 3 \end{array}\right] \] Select an Answer 3. \[ \left[\begin{array}{c} -1 \\ 2 \\ 1 \end{array}\right],\left[\begin{array}{c} -4 \\ 8 \\ 3 \end{array}\right] \] Select an Answer 4. \[ \left[\begin{array}{c} -3 \\ 2 \\ 1 \end{array}\right],\left[\begin{array}{c} 6 \\ -4 \\ -3 \end{array}\right],\left[\begin{array}{c} 6 \\ -4 \\ -4 \end{array}\right],\left[\begin{array}{r} -21 \\ 14 \\ 11 \end{array}\right] \]
Solución de tutoría real
Responder
Solución
Revisado y aprobado por el equipo de tutoría de UpStudy
Explicar 
Simplifique esta solución Mind Expander
To determine if the given sets of vectors span \( \mathbb{R}^{3} \), we must check if they are linearly independent and if they form a basis for \(\mathbb{R}^{3\). 1. For the first set, we can write the vectors as: \[ \begin{pmatrix} 2 & 2 & 0 \\ -1 & -1 & 0 \\ -2 & -3 & 1 \end{pmatrix} \] The presence of the third vector allows us to fill the third dimension, so yes, this set spans \(\mathbb{R}^{3}\). 2. In the second set, note that the third vector can be expressed as a combination of the first two: \[ 6 \vec{v_1} - 3 \vec{v_2} \] which means they aren't independent, hence they do not span \(\mathbb{R}^{3}\). 3. The third set only has two vectors, meaning it cannot span a three-dimensional space like \(\mathbb{R}^{3}\). 4. In the final set, you can notice that one vector is a linear combination of the others, thus they will not span \(\mathbb{R}^{3}\). In conclusion: - The first set spans \(\mathbb{R}^{3}\). - The second, third, and fourth sets do not span \(\mathbb{R}^{3}\). Now for some extra knowledge! Vectors are like characters in a story—they need depth and variety to create a rich narrative! When you're trying to span \( \mathbb{R}^{3}\), think of it like painting a wall: you can't just use one color; you need a palette that includes multiple shades. Feeling a bit lost with the concept of linear independence? Imagine you’re at a party and want to bring friends from different groups! If all your friends know each other (same linear combination), you’ll end up wearing the same outfit as them (linearly dependent)! But if you grab diverse friends, you’re bringing unique flair to the gathering (linearly independent)!
